# std::cyl_bessel_i, std::cyl_bessel_if, std::cyl_bessel_il

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Defined in header `<cmath>` |  |
| (1) |
float cyl_bessel_i ( float nu, float x ); double cyl_bessel_i ( double nu, double x ); long double cyl_bessel_i ( long double nu, long double x ); |  | (since C++17) (until C++23)
/* floating-point-type */ cyl_bessel_i( /* floating-point-type */ nu, /* floating-point-type */ x ); |  | (since C++23)
float cyl_bessel_if( float nu, float x ); | (2) | (since C++17)
long double cyl_bessel_il( long double nu, long double x ); | (3) | (since C++17)
Additional overloads |  |
Defined in header `<cmath>` |  |
template< class Arithmetic1, class Arithmetic2 > /* common-floating-point-type */ cyl_bessel_i( Arithmetic1 nu, Arithmetic2 x ); | (A) | (since C++17)

1-3) Computes the regular modified cylindrical Bessel function of nu and x. The library provides overloads of `std::cyl_bessel_i` for all cv-unqualified floating-point types as the type of the parameters nu and x.(since C++23)
A) Additional overloads are provided for all other combinations of arithmetic types.

### Parameters

nu | - | the order of the function
x | - | the argument of the function

### Return value

If no errors occur, value of the regular modified cylindrical Bessel function of nu and x, that is Inu(x) = Σ∞
k=0

(x/2)nu+2k
k!Γ(nu+k+1)

(for x≥0), is returned.

### Error handling

Errors may be reported as specified in math_errhandling.

- If the argument is NaN, NaN is returned and domain error is not reported.
- If nu≥128, the behavior is implementation-defined.

### Notes

Implementations that do not support C++17, but support ISO 29124:2010, provide this function if `__STDCPP_MATH_SPEC_FUNCS__` is defined by the implementation to a value at least 201003L and if the user defines `__STDCPP_WANT_MATH_SPEC_FUNCS__` before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the header `tr1/cmath` and namespace `std::tr1`.

An implementation of this function is also available in boost.math.

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their first argument num1 and second argument num2:

If num1 or num2 has type long double, then std::cyl_bessel_i(num1, num2) has the same effect as std::cyl_bessel_i(static_cast<long double>(num1), static_cast<long double>(num2)). Otherwise, if num1 and/or num2 has type double or an integer type, then std::cyl_bessel_i(num1, num2) has the same effect as std::cyl_bessel_i(static_cast<double>(num1), static_cast<double>(num2)). Otherwise, if num1 or num2 has type float, then std::cyl_bessel_i(num1, num2) has the same effect as std::cyl_bessel_i(static_cast<float>(num1), static_cast<float>(num2)). | (until C++23)
If num1 and num2 have arithmetic types, then std::cyl_bessel_i(num1, num2) has the same effect as std::cyl_bessel_i(static_cast</* common-floating-point-type */>(num1), static_cast</* common-floating-point-type */>(num2)), where /* common-floating-point-type */ is the floating-point type with the greatest floating-point conversion rank and greatest floating-point conversion subrank between the types of num1 and num2, arguments of integer type are considered to have the same floating-point conversion rank as double. If no such floating-point type with the greatest rank and subrank exists, then overload resolution does not result in a usable candidate from the overloads provided. | (since C++23)

### Example

Run this code

    #include <cmath>
    #include <iostream>

    int main()
    {
        // spot check for nu == 0
        const double x = 1.2345;
        std::cout << "I_0(" << x << ") = " << std::cyl_bessel_i(0, x) << '\n';

        // series expansion for I_0
        double fct = 1;
        double sum = 0;
        for (int k = 0; k < 5; fct *= ++k)
        {
            sum += std::pow(x / 2, 2 * k) / std::pow(fct, 2);
            std::cout << "sum = " << sum << '\n';
        }
    }

Output:

    I_0(1.2345) = 1.41886
    sum = 1
    sum = 1.381
    sum = 1.41729
    sum = 1.41882
    sum = 1.41886

### See also

cyl_bessel_jcyl_bessel_jfcyl_bessel_jl (C++17)(C++17)(C++17) | cylindrical Bessel functions (of the first kind) (function)

### External links

Weisstein, Eric W. "Modified Bessel Function of the First Kind." From MathWorld — A Wolfram Web Resource.

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